Hoeffding decomposition in $$H^1$$ spaces

The well known result of Bourgain and Kwapień states that the projection $$P_{\le m}$$ P ≤ m onto the subspace of the Hilbert space $$L^2\left( \Omega ^\infty \right) $$ L 2 Ω ∞ spanned by functions dependent on at most m variables is bounded in $$L^p$$ L p with norm $$\le c_p^m$$ ≤ c p m for $$1<p<\infty $$ 1 < p < ∞ . We will be concerned with two kinds of endpoint estimates. We prove that $$P_{\le m}$$ P ≤ m is bounded on the space $$H^1\left( {\mathbb {D}}^\infty \right) $$ H 1 D ∞ of functions in $$L^1\left( {\mathbb {T}}^\infty \right) $$ L 1 T ∞ analytic in each variable. We also prove that $$P_{\le 2}$$ P ≤ 2 is bounded on the martingale Hardy space associated with a natural double-indexed filtration and, more generally, we exhibit a multiple indexed martingale Hardy space which contains $$H^1\left( {\mathbb {D}}^\infty \right) $$ H 1 D ∞ as a subspace and $$P_{\le m}$$ P ≤ m is bounded on it.